The move comes across as the latest in a series of coups for the Canaries, with Redmond a highly-rated young winger who has impressed in the Championship in recent seasons.
Redmond will be reuniting with his former manager Chris Hughton, who said: "Nathan is a young and exciting player that I had the pleasure of working with at Birmingham. He is a player that was of interest to us for some time and the situation has now come about that we can get him on our terms."
Speaking of the deal, the player himself told the club's official website: "I'm very happy. It looks like a fantastic club, I'm just looking forward to getting started now. I know that there is only one football club in this city and it has an excellent fan base."
Redmond is Norwich's third signing of the season, after they completed deals for Javier Garrido and Ricky van Wolfswinkel.